Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
Predictive Control Design Based on Neural
Model of a Non-linear System
Anna Jadlovská, Nikola Kabakov, Ján Sarnovský
Department of Cybernetics and Artificial Intelligence, Faculty of Electrical
Engineering and Informatics, Technical University of Košice, Slovak Republic
anna.jadlovska@tuke.sk, nikola.kabakov@gmail.com, jan.sarnovsky@tuke.sk
Abstract: This paper provides two approaches for design of Generalized Predictive Control
(GPC) algorithm for a non-linear dynamic system. In the classical approach of the GPC
strategy the recursive method of least square is considered for calculating of the linearized
model parameters from the known analytic description of a particular non-linear system.
The other purpose of this paper is to show an intelligent approach in which a feed-forward
neural network (Multi Layer Perceptron – MLP) is used for modeling the same non-linear
system within the frames of the predictive control. The possibility of on-line estimation of
an actual parameters from the off-line trained neural model of the non-linear system by
means method of the instantaneous linearization in each sample point is considered in GPC
algorithm design. The validity of classical and neural GPC strategy is tested by computer
simulations in Matlab/Simulink language using architecture of S-functions of the library
PredicLib.
Keywords: non-linear dynamic system, neural model, parametric estimation, predictive
control algorithm
1
Introduction
Model Predictive Control (MPC) is one of the most wide spread advanced control
techniques for dynamic systems in the industry. The main idea of MPC algorithms
is to solve an optimization problem in order to find the control vector trajectory
that optimizes the cost function over a future prediction horizon [2], [3].
This paper provides two approaches for the design of Generalized Predictive
Control (GPC) algorithm for non-linear systems. In the classical approach of GPC
strategy (part 2) the recursive least square method (RLSM) is considered for the
calculation of the linearized model parameters from the known analytic
description of the dynamic system [8]. In the intelligent approach is a feedforward neural network (Multi Layer Perceptron – MLP) used for modeling and
the estimation of the actual parameters of the non-linear system which are
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
considered in GPC design (part 4-6). The neural model of non-linear system is
typically trained in advance, but the GPC controller is designed on-line using the
parameter estimation from the neural model. The main idea of this paper is to
show how on-line estimation of the actual parameters from off-line trained neural
model using the gain matrix is applied in the GPC algorithm [5]. The neural model
of non-linear system is linearized by means method of the instantaneous
linearization in each sample point and the result of this linearization technique –
the estimated parameters from neural ARX model (NARX) of the system are used
for design of the GPC algorithm (part 6). The practical simulations by the
language Matlab/Simulink, Neural Toolbox and the library PredicLib [6] of this
paper illustrate that the classical GPC and the neural GPC strategies using
linearization technique can be used for predictive control of a particular non-linear
dynamic system.
2
GPC Algorithm - Classical Approach
The calculating of the cost function for optimal control for the k-th step can be
considered as
J (k ) =
N2
Nu
j = N1
j =1
∑ e 2 ( k + j / k ) + ∑ λ Δu 2 ( k + j / k )
(1)
where for the predictive horizons holds N1 ≥ 1 , N2 ≥ N1, 1 ≤ Nu ≤ N2 , λ is the
positive weight coefficient and e(k + j / k ) = yˆ (k + j / k ) − r(k + j / k ) is the prediction
error, where yˆ( k + j / k ) is the predicted system output value and r ( k + j / k ) is
the system output required value. By [8] the most SISO (Single-Input/SingleOutput) systems when is considered the operation around a particular set-point
and after linearization can be described by the linear discrete AutoRegresive model
with eXternal input (ARX). Using the modification of ARX model we can get
CARIMA (Controlled AutoRegresive Integrate Moving Average) model which is
used in GPC algorithm [2] and is described by eq. (2)
A(q −1 ) y (k ) = B(q −1 )u (k − 1) +
C ( q −1 )
Δ
ξ (k )
(2)
where A(q-1), B(q-1), C(q-1) are the polynomials of the delay operator q-1:
A(q −1 ) = 1 + a1q −1 + a2 q −2 + ... + ana q − na , B(q −1 ) = b0 + b1q −1 + b2 q −2 + ... + bnb q − nb ,
C (q −1 ) = 1 + c1q −1 + c2 q −2 + ... + cnc q − nc
In the equation (2) ξ ( k ) is a white noise with zero mean and Δ = 1 − q −1 .
According [2] the future output value of the system is given by (3)
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Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
yˆ(k + j / k ) = Gj (q−1 )Δ u(k + j −1) + Γ j (q−1 )Δ u(k −1) + Fj (q−1 ) y(k )
(3)
In the equation (3) the polynomials G j ( q −1 ) , Γ j ( q −1 ) and F j ( q −1 ) are calculated
by solving of Diophantine equations:
C (q −1 ) = E j (q −1 ) A(q −1 )Δ + q − j F j (q −1 )
B (q −1 ) E j (q −1 ) = G j (q −1 )C (q −1 ) + q − j Γ j (q −1 )
The predictor (3) can be written as
yˆ = GΔu + F (q −1 ) y (k ) + Γ (q −1 )Δu (k − 1)
(4)
where
⎡ yˆ( k + N1 / k ) ⎤
⎢ yˆ( k + N + 1/ k ) ⎥
1
⎥,
yˆ = ⎢
⎢
⎥
M
⎢
⎥
ˆ(
y
k
+
N
/
k
)
2
⎣
⎦
⎡ g N1
⎢
g N +1
G = ⎢⎢ 1
M
⎢
⎢⎣ g N 2
Δu(k)
⎡
⎤
⎢ Δu(k +1) ⎥
⎥,
Δu = ⎢
⎢
⎥
M
⎢
⎥
Δu(k
+
N
-1)
u
⎣
⎦
g N1 −1 L g N1 − Nu +1 ⎤
⎡ Γ N1 (q -1 ) ⎤
⎡ FN1 ( q −1 ) ⎤
⎢
⎥
⎢
⎥
⎥
-1
−1
g N1 L g N1 − Nu + 2 ⎥
⎢ Γ N1 +1 (q )⎥
⎢ FN1 +1 ( q ) ⎥
-1
−1
, Γ (q ) = ⎢
⎥ , F (q ) = ⎢
⎥.
M
O
M ⎥
M
M
⎢
⎥
⎢
⎥
⎥
⎢⎣ Γ N 2 (q -1 ) ⎥⎦
⎢⎣ FN2 ( q −1 ) ⎥⎦
g N 2 −1 L g N2 − Nu +1 ⎥⎦
Because two last items of (4) depends only on the previous states, we can include
those into one item f, then the equation of the predictor is ŷ = GΔu + f. The cost
function (1) can be written in the matrix form for the computing of an optimal
control
J = (GΔu + f − r )T (GΔu + f − r ) + λ ΔuT Δu
(5)
where r = [ r ( k + N1 ), r ( k + N1 + 1),..., r ( k + N 2 )] is the reference trajectory and λ
T
is the weight coefficient.
The equation (5) can be written as
J=
1 T
Δu HΔu + bT Δu + f 0
2
(6)
where H = 2(G T G + λ I ) , bT = 2( f − r )T G , f 0 = ( f − r )T ( f − r ) .
The minimum of the cost function J can be found by making gradient of J equal
to zero, which leads to
Δu = − H −1b = −(G T G + λ I ) −1 G T ( f − r ) .
– 95 –
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
The result of this equation is the trajectory consisting from the increments of the
control signal and the first of them is applied on the system and is given by:
Δu (k ) = K (r − f ) , where K is the first row of the matrix (G T G + λ I ) −1 G T , so an
actual controller output value is u (k ) = u (k − 1) + K (r − f ) .
For the systems with the constrains on the controller output value, on the
controller increment output value or on the system output value, the vector Δu is
calculated by function quadprog of Optimization Toolbox of the language Matlab
Δu = quadprog (H, b T, LCON, v, UMIN, UMAX), where UMIN ≤ Δu ≤ UMAX , LCON Δu ≤ v.
The vectors UMIN and UMAX are the column vectors those elements are minimal and
maximal values of Δu(k). With using the matrix LCON and the vector v can be
defined the system of an inequalities which insures that constrain conditions will
be satisfied [6].
LCON 1
⎡1
⎢1
=⎢
⎢#
⎢
⎣1
LCON 2
⎡ −1 0
⎢ −1 −1
=⎢
⎢#
#
⎢
⎣ −1 −1
0
1
#
1
" 0⎤
⎡
⎡1⎤ ⎤
" 0 ⎥⎥
⎢
⎥
, v1 = ⎢( uMAX − u (k − 1) ) . ⎢⎢# ⎥⎥ ⎥
⎥
% #
⎢⎣
⎢⎣1⎥⎦ ⎥⎦
⎥
" 1⎦
0⎤
⎡
⎡ −1⎤ ⎤
0 ⎥⎥
⎢
⎥
, v2 = ⎢( uMIN − u (k − 1) ) . ⎢⎢ # ⎥⎥ ⎥
% #⎥
⎢⎣ −1⎥⎦ ⎦⎥
⎥
⎣⎢
" −1⎦
"
"
⎡
⎤
⎡1⎤
⎢
⎥
LCON 3 = G, v3 = ⎢ yMAX . ⎢⎢# ⎥⎥ − F (q −1 ). y (k ) − Γ (q −1 ).Δu (k − 1) ⎥
⎢⎣
⎥⎦
⎢⎣1⎥⎦
⎡
⎤
⎡ −1⎤
⎢
⎥
LCON 4 = −G, v4 = ⎢ yMIN . ⎢⎢ # ⎥⎥ + F (q −1 ). y (k ) + Γ (q −1 ).Δu (k − 1) ⎥
⎢⎣
⎥⎦
⎢⎣ −1⎥⎦
LCON
⎡ LCON 1 ⎤
⎡ v1 ⎤
⎢L ⎥
⎢v ⎥
= ⎢ CON 2 ⎥ , v = ⎢ 2 ⎥
⎢ v3 ⎥
⎢ LCON 3 ⎥
⎢ ⎥
⎢
⎥
L
⎣v4 ⎦
⎣ CON 4 ⎦
In the programmable environment Matlab/Simulink was designed GPC algorithm
for the dynamic non-linear SISO system using S-functions.
Algorithm GPC for the calculation of the control signal value for the k - th step:
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Acta Polytechnica Hungarica
1
Vol. 5, No. 4, 2008
reading of the polynomials A(q-1), B(q-1), C(q-1) of the linearized discrete
model (2) of the non-linear system, the reference vector trajectory r(k) and the
output of the system y(k),
2 cyclical calculating of the polynomials Gj(q-1), Γj (q-1) and F(q-1) by solving
Diophantine equations for j = 1, 2, .., N2,
3 creating of the matrixes G, Γ and F,
4 if is required the constrain for the values of u(k), Δu(k) or y(k), then continue
by the step 8,
5 calculating of the feedback gain of the control vector K,
6 calculating of the controller output increment Δu(k) = K (w – f),
7 continue by the step 12,
8 creating of the matrix LCON and vectors v, UMIN, UMAX,
9 calculating of the matrixes H and b T,
10 Δu = quadprog(H, b T, LCON, v, UMIN, UMAX),
11 Δu(k) is the first element of the vector Δu,
12 u(k) = u(k-1) + Δu(k), k = k+1 and continue by the step 1.
3
Verification of GPC Algorithm for a Non-linear
System by Simulation
The simulation model of the non-linear system consists of two tanks, one
cylindrical and the other spherical with the free outflow. The inflow to the first
tank is controlled by the relative open of the valve V1. The outflow from the first
tank is an inflow to the second one.
The physical variables and the parameters of the hydraulic system which is
described in Fig. 1 are:
h1(t) [m] - the water level in cylindrical tank,
h1MAX(t) [m] - the max water level in cylindrical tank,
h2(t) [m] - the water level in the spherical tank,
h2MAX(t) [m] - the max water level in the spherical tank,
M1 (t) [kg/s] - the mass inflow to the cylindrical tank,
M2 (t) [kg/s] - the mass inflow to the spherical tank,
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
M3 (t) [kg/s] - the mass outflow from the spherical tank,
MZ1 (t) [kg/s] - the disturbance mass inflow to the cylindrical tank,
MZ2(t) [kg/s] - the disturbance mass inflow to the spherical tank,
u1(t) - the rise of an input outlet of the cylindrical tank,
S [m2] - the tank’s bottom area,
S1 [m2] - the area of the outflow of the cylindrical tank,
R [m] - the diameter of the spherical tank, S2 [m2] is the outflow space of spherical tank.
Figure 1
Model of the hydraulic system
For changing the levels h1(t) and h2(t) according to [6] it holds that
dh1 (t )
= M 1 (t ) − M 2 (t ) + M Z 1 (t )
dt
dh (t )
ρπ (2 Rh2 ( t ) − h22 ( t )) 2 = M 2 (t ) − M 3 (t ) + M Z 2 (t )
dt
ρS
(8)
The mass flows M1(t), M2(t) and M3(t) can be described by the equations (9),
where kV1 is the constructing constant of the input outlet of the cylindrical tank and
f(u1(t)) is characteristic function of the valve V1
M 1 (t ) = kV 1 f (u1 (t )) M 1MAX , M 2 (t ) = ρ S1 2 gh1 (t ), M 3 (t ) = ρ S2 2 gh2 (t )
– 98 –
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Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
The mass inflow M1(t) depends on the relative open value of u1 (t ) ∈ 0,1 and on
the value of maximal mass inflow M1MAX. Let the function f(u1(t)) be linear. Then
the equation for the mass inflow can be written as
M 1 (t ) = kV 1u1 (t ) M 1MAX
(10)
After inducting equations (9) and (10) to the equations (8) we obtain non-linear
differential equations (11) describing dynamics of the changing levels in the
simulation model - two tanks without interaction
dh1 (t ) kV 1u1 (t ) M 1MAX − ρ S1 2 gh1 (t ) + M Z 1 (t )
=
dt
ρS
dh2 (t ) ρ S1 2 gh1 (t ) − ρ S2 2 gh2 (t ) + M Z 2 (t )
=
dt
ρπ (2 Rh2 (t ) − h22 (t ))
(11)
By expansion to Taylor series for the set-point SP = [u10, h10, h20] we obtain the
linearized model, which can be written by Laplace transformation as transfer
functions FH 2 / U1 ( s ) and then using Z-transformation can be obtained the discrete
transfer function of the dynamic system F ( z ) = B ( z ) / A ( z ) .
For verification of designed GPC algorithm was used the simulation language
Matlab/Simulink. The functional block of GPC controller is included into the
library PredicLib [6]. The parameters of the simulation model of the hydraulic
system are M1MAX = 500kg/s, h1MAX = 6m, S = 7.07m2, S1 = 0.0314m2, R = 1.5m,
S2 = 0.0314 m2 and ρ = 1000 kg/m3.
For the calculation of the linearized model parameters the recursive method of the
least square was used for the set-point SP that corresponds to the actual value of
the system output h2(t). After conversion of the discretized transfer function to
time area operator z is substituted with operator q. By this way the polynomials
A(q-1) and B(q-1) of the discrete CARIMA model (2) can be acquired. The
polynomial C(q-1) is time invariant and is chosen to be equal to one. The orders of
discrete model´s polynomials are na = 2, nb = 2 and nc = 0.
The prediction horizon for the output of the system is chosen on ten steps forward
and for the control signal on five steps, so N1 = 1, N2 = 10, Nu = 5. The sample
period is TVZ = 10s.
Next is necessary to consider the constrains for the control signal value and the
system output value because u1 (t ) ∈ 0,1 and h2 (t ) ∈ 0,3 . The weight
coefficient for the control signal value increment Δu(t) is λ=0.1.
The control scheme of the simulation in an environment Matlab/Simulink is
described in Fig. 2.
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
[W(:,1),W(:,3:12)*2]
ref(t)
y(t)
Vector of reference
trajectory
A(z)
u(t)
B(z)
1
u1(t)
C(z)
h1(t)
h2(t)
Mz2(t)
Adaptive GPC Disturbance Mz1
SISO
Mz2(t)
C(z)
h1(t)
Mz1(t)
Mz1(t)
Coupled tanks,
one cylindrical and
one spherical
Disturbance Mz2
Mux
Sum
Mux
u(t),h2(t),ref(t)
-KGain
Band-Limited
White Noise
B(z)
h2(t)
A(z)
[w(:,1),w(:,2)*2]
Reference value
Figure 2
The simulation predictive control scheme – classical approach
On the output of the system have effect disturbance in the form of the white noise
that simulates the measurement error. Also the disturbance mass inflows MZ1 and
MZ2 act on the system.
The disturbance mass inflow to the cylindrical tank has value MZ1 = 50kg.s-1 and
acts on in time from 2500s to 3500s. The disturbance mass inflow to the spherical
tank have value MZ2 = 20kg.s-1 and acts on in time from 6500s to 7500s.
The result of tracking of the reference trajectory ref(t) by the system’s output h2(t)
using GPC algorithm is on Fig. 3. The output of the GPC controller – the optimal
control signal u1(t) is described in Fig. 4.
2
1.8
ref(t)
h2(t)
1.6
1.4
1
2
h (t) [m]
1.2
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
t [s]
6000
7000
8000
9000
10000
Figure 3
Tracking the reference trajectory ref(t) by the output of the non-linear system h2(t)
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Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
1
0.9
0.8
0.7
u(t)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
4000
5000
t [s]
6000
7000
8000
9000
10000
Figure 4
The GPC controller output – control signal u1(t)
4
Predictive Control Structure Using Parameter
Estimation from a Neural Model
Next we will consider about the GPC algorithm design for a non-linear system
(hydraulic two tanks system) using the neural model of NARX structure [2], [7].
The predictive control scheme using the estimation the system parameters from
off-line trained neural NARX model, which are applied in the algorithm of GPC,
is illustrated on Fig. 5.
estimated parameters
ai(k), bi(k)
Prediction output
of the process
yref(k)
GPC
controller
Neural model
of process
u(k)
y(k)
Process
Figure 5
The predictive control scheme based on the neural model - intelligent approach
The loop consists of the controlled non-linear hydraulic system described by the
equations (11), the neural process model of NARX structure and GPC controller.
Because GPC algorithm needs parameters of a linear model of the dynamic
system, in this paper a linear model is extracted from the non-linear neural NARX
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
model by calculation of the gain matrix (part 5). The estimated parameters of the
dynamic system from off-line trained neural model are applied for the calculation
of the predictor by algebraic theory which is used in GPC algorithm [5], [6]. The
optimal predictive controller output vector Δuopt is determined by minimization of
the cost function (1). The first term in the cost function refers to the square
variation of the predicted system output from the desired reference trajectory,
while the second term is added in order to limit the controller output, greater λ
yields less active the controller output. The first element of the calculated optimal
controller output vector is directed to the hydraulic system input. The remaining
vector elements are not utilized and the entire procedure is repeated at the time
t = ( k + 1) TVZ , (the principle of receding horizon) [3]. The first prediction horizon
N1 is usually chosen to be 1. The choice of the second prediction horizon N 2 and
the control horizon N u are: N 2 is usually chosen as to cover the most of the
control system’s transient, while N u , which denotes the significance of the future
controller outputs shouldn’t be greater then N 2 / 2 .
5
Neural Model of a Non-linear Dynamic System
In this part we will discuss some basic aspects of the non-linear system
identification using from among numerous neural networks structures only MultiLayer Perceptron – MLP (a feed-forward neural network) [1], [9] with respect to
the model based neural predictive control, where the control law is based upon the
neural model. We will use in this paper a feed-forward neural network MLP with
a single hidden layer. This structure is shown in the matrix notation in Fig. 6 [5],
[7]. The matrix W1 represents the input weights, the matrix W2 represents the
output weights, Fh represents a vector function containing the non-linear (tanh)
neuron functions. The “1” shown in Fig. 6 together with the last column in the
matrix W1 gives the offset in the network. The network input is represented by
)
vector Z in and the network output is represented by vector Z out . The mismatch
)
between the desired output Z out and an aproximated output Z out is the prediction
error E .
The output from the neural network MLP can be written as
⎛ ⎡Y ⎤ ⎞
X 2 = W2 Fh ⎜ W1 ⎢ 0 ⎥ ⎟ .
⎝ ⎣ 1 ⎦⎠
(12)
From a trained MLP by Back-Propagation Error Algorithm (BPA – the first-order
gradient method) a gain matrix M can be found by differentiating with respect to
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Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
the input vector of the network. The gain matrix M can be calculated using (12)
as
M=
dZˆ out dX 2 dX 2 dY1 dX1
=
=
. T . T = W2 .Fh´ ( X 1 ) .W1*
T
T
dZ in
dYo
dY1 dX1 dYo
where W1*
(13)
W1 (excluding the last column).
The above mentioned the gain matrix M allows an on-line estimation of the
actual parameters from an off-line trained neural model of the non-linear system.
Zin
X1
Y0
1
Fh
Zout
X2 Zˆ out
Y1
W1
W2
+E
-
Figure 6
The matrix block diagram of the neural network MLP
With the inspiration from linear ARX model described in [8], next we will use a
general model structure suitable for representing the dynamics of a wide range of
non-linear system – neural ARX model (NARX) [4], [10], which is defined by
Yˆ (k ) = FN (Y (k − 1),...,Y (k − p ),U (k − 1),...,U (k − m ),θ )
Y (k ) = Yˆ (k ) + E (k )
(14)
where FN is the unknown non-linear vector function to be approximated, E(k) is
the prediction error, p and m denote the number of delayed outputs and inputs. The
non-linear mapping FN can be approximated by a feed-forward neural network,
e.g. MLP. The feed-forward neural network FN is configured to represent the
NARX model by applying p delayed values of the system output and m delayed
values of the system input to the network inputs and assigning it’s output Ŷ(k) to
be Y(k), θ is the vector of the network parameters.
An optimal value of the network parameters is usually obtained by using a training
algorithm BPA that minimizes the following cost function:
N
(
J NN (θ ) = ∑ Y ( i ) − Yˆ ( i, θ )
i =1
)
2
(15)
on the basis of this gradient with respect to the network parameters. N in
equation (15) is the length of input-output data set used for the network training.
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
We will consider that the neural NARX model has the input vector Zin and the
output vector Ẑ out :
Zin (k ) = {Y (k −1),K,Y (k − p),K, U (k −1),K, U (k − m)}
Zˆ (k ) = Yˆ (k )
(16)
out
After training the neural network MLP by BPA the actual gain matrix M(k) can
be on-line estimated and calculated for neural ARX model as
M(k) =
dZˆ out ( k )
dZinT ( k )
{
=
dYˆ ( k )
d {Y ( k −1) ............ U ( k − m)
= −aˆ1 ( k ) K − aˆ p ( k ) bˆ1 ( k ) K bˆm ( k )
}
}
T
=
(17)
where âi (k ) for i=1,…, p, b̂i (k ) for i = 1, …, m are estimated parameters of
the neural NARX model for step k . Using the method of an instantaneous
linearization we can extract the parameters of the dynamic system from an off-line
trained neural model using the gain matrix [5], [6] instead an estimation
parameters of the linear model by the method of the least square. The GPC
algorithm uses these estimated parameters from the neural model for control
signal calculation.
6
Simulation Results of GPC Algorithm based on
Neural Model – Intelligent Approach
The results of the estimation the system parameters from an off-line trained neural
NARX model and their application in GPC algorithm using an algebraic theory
are presented for a non-linear test SISO system – two tanks system (11). The
output of the system is the water level in the second tank y (t ) = h2 (t ). . We consider
NARX model with 4 inputs and 6 neurons in the hidden layer. The activation
functions in the hidden layer are „tanh“ functions and in the output layer is
selected a linear function. The actual values of the estimated parameters can be
obtained from the gain matrix M (k ) by (17).
The GPC algorithm in the k - th step consists the following calculations:
1 the real output of the system (11) y (k ) is measured,
2 the calculation of the linearized parameters âi ,b̂i from the neural NARX
model by the gain matrix (17),
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Acta Polytechnica Hungarica
Vol. 5, No. 4, 2008
3 the estimated parameters are implemented in the GPC algorithm and an
optimal control value is obtained by minimized the cost function (1),
4 an optimal control value is applied on the system input,
5 k = (k+1)TVZ and go to step 1.
The implementation of the resulting GPC controller including a neural network
model for parameter estimation is shown on Fig. 7.
[W(:,1),W(:,3:12)*2]
Vector of referenc e
trajectory
1
ref (t )
y (t )
A(z)
u 1(t )
u (t )
B(z)
Mz1(t)
C(z)
Adapti ve GPC Disturb ance M z1
SISO
Mz2(t)
C(z)
h1(t)
h1(t)
Mz1(t )
Di sturbance Mz 2
Mz2(t )
h2(t)
Coupl ed tanks,
one c yli ndri cal and
one spheri cal
Mux
Sum
Mux
u(t),h2(t),re f(t)
-KBand-Li mi ted
White Noi se
Gain
B(z)
A(z)
y (t )
N ETin
N ETin(t)
u (t )
Extracting of ARX m odel
parameters from NARX
NET in
[w(:,1),w(:,2)*2]
Referenc e val ue
Figure 7
The GPC control scheme using neural model
The presentation of the results of GPC strategy using on-line parameter estimation
from an off-line trained neural model is illustrated on Figs. 8 and 9, where the
noise of measurement – the white noise is added to the output of the system (11) at
N1 = 1, N 2 = 10 , N u = 5, λ = 0.1 . The disturbances input flows are added to the
input of the non-linear system.
The plot on Fig. 8 compares the reference output of the system h2ref(t) and the
actual output of the closed-loop system h2(t). A perfect model-following behaviour
is achieved, although we can see an oscillating control signal on Fig. 9. This
simulation example shows the possibility of an application of the neural modeling
using the structure ARX known from the theory of the linear identification and
also the possibility to apply GPC algorithm known from linear control theory for
the control of non-linear SISO systems which have no the hard nonlinearities in
the control structure in Fig. 5.
– 105 –
A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
2.5
ref(t)
h 2 (t)
ref(t),h2(t) [m]
2
1.5
1
0.5
0
0
1000
2000
3000
4000
5000
t[s]
6000
7000
8000
9000
10000
Figure 8
Tracking the reference trajectory ref(t) by the output of the non-linear system h2(t) using the parameter
estimation from neural NARX model
0 .7
0 .6
0 .5
u(t)
0 .4
0 .3
0 .2
0 .1
0
0
1000
2000
3000
4000
5000
t[s ]
6000
7000
8000
9000
10000
Figure 9
The control signal of the GPC controller based on parameter estimation from neural NARX model
The library PredicLib is created as a software tools by Simulink and Matlab
language built-in functions. It contains blocks of Model Predictive Control (MPC)
algorithms, concrete GPC algorithms for SISO and MIMO systems and MPC
algorithms based on the state space model of the systems. PredicLib (Fig. 10)
contains also the blocks for an extracting of the parameters from neural NARX
model and extracting of the discrete state space model parameters from neural
Nonlinear Innovation State Space (NISS) model of the controlled system, a block
of the discrete Kalman estimator and a block for conversion of the discrete state
space model to the transfer function. Some demo simulations of MPC algorithms
for the control of the linear and the non-linear systems are included in the library.
Some of the predictive control algorithms blocks demo applications created by
using Matlab Web Server are accessible Virtual laboratory CyberVirtLab
[http://cyberneticsmws.fei.tuke.sk/MatlabWebServer_welcome].
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Acta Polytechnica Hungarica
ref (t)
y (t)
Vol. 5, No. 4, 2008
ref (t)
u(t)
y (t)
GPC
SISO
GPC
MIMO
ref (t)
ref (t)
y (t)
y (t)
A(t)
u(t)
u(t)
ref (t)
x(t)
u(k)
A(z)
MPC based on
state space model Extracting of ARX model
parameters from NARX
ref (t)
u(t)
A(t)
B(t)
B(t)
B(t)
C(t)
C(t)
C(t)
GPC
MIMO
Adaptive
D(t)
GPC
SISO
Adaptive
NETin
NETin
u(t)
C(t)
A(t)
Extracting of DSS model
parameters from NISS
xe(k)
Discrete
Kalman
estimator
D(t)
B(t)
x(t)
A(t)
y (k)
B(t)
u(t)
B(t)
A(t)
num(z)
den(z)
Transformation of
DSS to T F
MPC based on
state space model
Adaptive
GPC_SISO_DISC
GPC_SISO_DISC_A
GPC_SISO_CON
GPC_SISO_CON_A
GPC_MIMO_DISC
GPC_MIMO_DISC_A
GPC_MIMO_CON
GPC_MIMO_CON_A
MPC_SISO_DISC
MPC_SISO_DISC_A
MPC_SISO_CON
MPC_SISO_CON_A
GPC_SISO_ILIN
GPC_SISO_RMNS
GPC_SISO_NARX
GPC_SISO_NISS
Trainig data
Verification
Figure 10
Screenshot of the library PredicLib from Simulink
Conclusion
In this paper are presented two approaches - classical and intelligent for the design
of GPC algorithm for a non-linear system. The disadvantage of the classical
approach of GPC strategy using as linearization technique for an estimate
parameters of the dynamic model is that analytic description of non-linear system
must be given. If analytical model of the dynamic system is unknown the better
solution is to use an intelligent approach – GPC based on the neural NARX or
NISS model for MIMO non-linear systems. In this paper is neural NARX model
trained as an one-step predictor for a non-linear SISO system. After training this
NARX model was used for on-line estimation of the system parameters which
allow to calculate a linear predictor. This linear predictor of the system was used
for solving of an optimization problem of GPC algorithm.
The practical simulations by the language Matlab/Simulink, Neural Toolbox and
PredicLib illustrate, that this intelligent neural GPC control strategy using
linearization technique by the gain matrix produces more excellent performance
for control of the non-linear system as GPC strategy using of the classical
approach.
Applying the principle of an instantaneous linearization to the GPC design gives
tremendous advantages over the conventional nonlinear predictive control design.
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A. Jadlovská et al.
Predictive Control Design Based on Neural Model of a Non-linear System
Acknowledgment
This research has been supported by the Scientific Grant Agency of Slovak
Republic under project Vega No. 1/0617/08 Multiagent Network Control Systems
with Automatic Reconfiguration.
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